Triple

T3214179
Position Surface form Disambiguated ID Type / Status
Subject Fibonacci sequence E67350 entity
Predicate hasClosedForm P46241 FINISHED
Object Binet’s formula
Binet’s formula is the closed-form expression that computes the nth Fibonacci number using powers of the golden ratio and its conjugate.
E67350 NE FINISHED

How this triple was built (5 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: Binet’s formula | Statement: [Fibonacci sequence, hasClosedForm, Binet’s formula]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Binet’s formula
Context triple: [Fibonacci sequence, hasClosedForm, Binet’s formula]
  • A. Fibonacci sequence
    The Fibonacci sequence is an infinite series of numbers where each term is the sum of the two preceding ones, widely used in mathematics, art, and design due to its connection with the golden ratio and natural growth patterns.
  • B. Bernoulli numbers
    Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
  • C. Pascal's triangle
    Pascal's triangle is a triangular array of numbers in which each entry is the sum of the two directly above it, widely used in combinatorics, algebra, and probability.
  • D. Euler’s theorem
    Euler’s theorem is a fundamental result in number theory stating that for any integer a coprime to n, a raised to the power of φ(n) is congruent to 1 modulo n.
  • E. Euler–Maclaurin summation formula
    The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Binet’s formula
Triple: [Fibonacci sequence, hasClosedForm, Binet’s formula]
Generated description
Binet’s formula is the closed-form expression that computes the nth Fibonacci number using powers of the golden ratio and its conjugate.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Binet’s formula
Target entity description: Binet’s formula is the closed-form expression that computes the nth Fibonacci number using powers of the golden ratio and its conjugate.
  • A. Fibonacci sequence chosen
    The Fibonacci sequence is an infinite series of numbers where each term is the sum of the two preceding ones, widely used in mathematics, art, and design due to its connection with the golden ratio and natural growth patterns.
  • B. Bernoulli numbers
    Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
  • C. Pascal's triangle
    Pascal's triangle is a triangular array of numbers in which each entry is the sum of the two directly above it, widely used in combinatorics, algebra, and probability.
  • D. Euler’s theorem
    Euler’s theorem is a fundamental result in number theory stating that for any integer a coprime to n, a raised to the power of φ(n) is congruent to 1 modulo n.
  • E. Euler–Maclaurin summation formula
    The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
  • F. None of above.
PD Predicate disambiguation gpt-5-mini-2025-08-07
Target predicate: hasClosedForm
Context triple: [Fibonacci sequence, hasClosedForm, Binet’s formula]
  • A. hasSymbolicForm
    Indicates that one entity serves as the symbolic representation or abstract form of another entity.
  • B. hasFullForm
    Indicates that one entity is the complete, expanded, or unabbreviated form of another entity.
  • C. admitsSolution
    Indicates that a problem, system, or situation allows for or possesses at least one valid solution.
  • D. hasFormulation
    Indicates that one entity is expressed, prepared, or configured in a particular form or composition defined by another entity.
  • E. hasNonStandardForm
    Indicates that an entity possesses a form, variant, or representation that deviates from the standard, canonical, or commonly accepted form.
  • F. None of above. chosen

Provenance (7 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69ad858ac36c81909962589cd277d6e2 completed March 8, 2026, 2:19 p.m.
NER Named-entity recognition batch_69adaabd01d48190be0dc610b9987a25 completed March 8, 2026, 4:58 p.m.
NED1 Entity disambiguation (via context triple) batch_69b26237211c81908cddc4e8d42e497e completed March 12, 2026, 6:50 a.m.
NEDg Description generation batch_69b264c54c508190be85da879935e7f4 completed March 12, 2026, 7:01 a.m.
NED2 Entity disambiguation (via description) batch_69b268d60bd0819097194da2e9065947 completed March 12, 2026, 7:18 a.m.
PD Predicate disambiguation batch_69ad9e09b83881908801d79c3d9254f9 completed March 8, 2026, 4:04 p.m.
PDg Predicate description generation batch_69ada0f9259c8190afbc5ad0fa55436b completed March 8, 2026, 4:16 p.m.
Created at: March 8, 2026, 3:07 p.m.